%I #14 Jun 24 2023 16:03:06
%S 1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,3,3,1,1,1,2,4,4,3,1,1,1,2,4,6,5,
%T 4,1,1,1,2,4,7,8,7,4,1,1,1,2,4,8,10,12,8,5,1,1,1,2,4,9,13,15,16,10,5,
%U 1,1,1,2,4,9,15,20,22,21,12,6,1,1,1,2,4,9,17,25,31,31,27,14,6,1,1,1,2,4,9
%N Triangle read by rows, such that T(n,k) equals the k-th term of the convolution of the two prior rows indexed by (n-k) and (k-1).
%C Row sums are A091501. Convergent of rows is A091500: {1, 1, 2, 4, 9, 20, 47, 113, 275, 676, 1685, 4271, 10843, 27801, 71611, 185795, ...}.
%H Paul D. Hanna, <a href="/A091499/b091499.txt">Table of n, a(n) for n = 0..1652</a>
%F T(n, k) = Sum_{j=0..min(n-k, k)} T(n-k, j)*T(k-1, k-j-1), with T(n, 0) = 1, T(n, n) = 1 for n >= 0.
%e T(7,3) = 4 = third term of convolution[{1,1,2,2,1},{1,1,1}] = 1*1 + 1*1 + 2*1 = T(4,0)*T(2,2) + T(4,1)*T(2,1) + T(4,2)*T(2,0).
%e T(14,5) = 19 = 5th term of convolution[{1,1,2,4,8,10,12,8,5,1},{1,1,2,2,1}] = 1*1 + 1*2 + 2*2 + 4*1 + 8*1 = T(9,0)*T(4,4) + T(9,1)*T(4,3) + T(9,2)*T(4,2) + T(9,3)*T(4,1) + T(9,4)*T(4,0).
%e Rows begin:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 1, 2, 1;
%e 1, 1, 2, 2, 1;
%e 1, 1, 2, 3, 3, 1;
%e 1, 1, 2, 4, 4, 3, 1;
%e 1, 1, 2, 4, 6, 5, 4, 1;
%e 1, 1, 2, 4, 7, 8, 7, 4, 1;
%e 1, 1, 2, 4, 8, 10, 12, 8, 5, 1;
%e 1, 1, 2, 4, 9, 13, 15, 16, 10, 5, 1;
%e 1, 1, 2, 4, 9, 15, 20, 22, 21, 12, 6, 1;
%e 1, 1, 2, 4, 9, 17, 25, 31, 31, 27, 14, 6, 1;
%e ...
%o (PARI) {T(n,k)=if(k>n||n<0||k<0,0,if(k<=1||k==n,1, sum(j=0,min(n-k,k),T(n-k,j)*T(k-1,k-j-1))))}
%o (PARI) \* Faster - using N X N matrix in recursion *\
%o N = 30; M=matrix(N,N);
%o {T(n, k) = M[n+1,k+1] = if(k>n||n<0||k<0, 0, if(k<=1||k==n, 1, sum(j=0, min(n-k, k), T(n-k, j)*if(k<1||k-j<1,0,M[k, k-j]))))}
%o for(n=0,N-1,for(k=0,n,print1(T(n,k),","));print(""))
%Y Cf. A091500, A091501.
%K nonn,tabl
%O 0,9
%A _Paul D. Hanna_, Jan 16 2004